Matrices That Are Self-Congruent Only Via Matrices of Determinant

arXiv:1004.3603v1 [math.RT] 21 Apr 2010 20)1620-1632. (2009) ∗ hsi h uhr eso fawr htwspbihdin published was that work a of version authors the is This arcsta r efcnretol via only self-congruent are that Matrices nte ro fti rtro over criterion this of proof another aoia ar fsmercadse-ymti arcsf matrices skew-symmetric Let and symmetric gruence. of pairs canonical aedtriat1i n nyif only and if 1 determinant have 2 aeaohrpofo hscieinover criterion this of proof consi another [ fo also gave bilinear Johnson was of and classification 2 Dopico, Riehm’s characteristic ley, on of based is case proof (the Their dimension odd of 8326]cniee etrspace vector a considered 2853–2863] hypoe htalioere of isometries all that proved They atLk iy th813 [email protected] 84103, Utah City, Lake Salt ooicadSehmn [ Szechtman, Docović and - eateto ahmtc,Uiest fUtah of University Mathematics, of Department nttt fMteais eehhnisa3 Tereshchenkivska Mathematics, of Institute nttt fMteais eehhnisa3 Tereshchenkivska Mathematics, of Institute iv kan,[email protected] Ukraine, Kiev, iv kan,[email protected] Ukraine, Kiev, arcso eemnn one determinant of matrices M etemti fteblna omon form bilinear the of matrix the be ayn .Gerasimova G. Tatyana ldmrV Sergeichuk V. Vladimir oe .Horn A. Roger ierAgbaAppl. Algebra Linear Abstract rc mr ah Soc. Math. Amer. Proc. V 1 vrafield a over V V F a ootooa summands orthogonal no has noe ihablna form. bilinear a with endowed sn u aoia matrices canonical our using R and F C 2 20)796–813] (2008) 428 fcaatrsi not characteristic of sn Thompson’s using ierAgbaAppl. Algebra Linear V m.Coak- rms. 3 (2005) 133 egive We . ∗ rcon- or dered). 431 for congruence and obtain necessary and sufficient conditions involv- ing canonical forms of M for congruence, of (M T , M) for equivalence, and of M −T M (if M is nonsingular) for similarity. AMS classification: 15A21; 15A63 Keywords: Canonical forms; Congruence; Orthogonal groups; Symplectic matrices

1 Introduction

Fundamental results obtained by Docovi´cand- Szechtman [4] lead to a description of all n-by-n matrices M over any ﬁeld F such that

S nonsingular and ST MS = M imply det S =1. (1)

Over a field of characteristic not 2, we give another proof of their description and obtain necessary and sufficient conditions on M that ensure (1) and involve canonical forms of M for congruence, of (M T , M) for equivalence, and of M −T M (if M is nonsingular) for similarity. Of course, if F has characteristic 2 then every nonsingular matrix M satisfies (1). A vector space V over F endowed with a bilinear form B : V V F is called a bilinear space. A linear bijection : V V is called an×isometry→ if A → B( x, y)= B(x, y) for all x, y V. A A ∈ If B is given by a matrix M, then the condition (1) ensures that each isometry has determinant 1; that is, the isometry group is contained in the special linear group. A bilinear space V is called symplectic if B is a nondegenerate skew- symmetric form. It is known that each isometry of a symplectic space has determinant 1 [1, Theorem 3.25]. If B is given by the matrix

0 Im Z2m := , (2) Im 0 − then each isometry is given by a symplectic matrix (a matrix S is symplectic T if S Z2mS = Z2m), and so each symplectic matrix has determinant 1. We denote by Mn(F) the set of n n matrices over a field F and say that × A, B Mn(F) are congruent if there is a nonsingular S Mn(F) such that T ∈ −1 ∈ S AS = B; they are similar if S AS = B for some nonsingular S Mn(F). ∈ 2 The following theorem is a consequence of Docovićand- Szechtman’s main theorem [4, Theorem 4.6], which is based on Riehm’s classification of bilinear forms [10].

Theorem 1. Let M be a square matrix over a ﬁeld F of characteristic different from 2. The following conditions are equivalent:

(i) M satisﬁes (1) (i.e., each isometry on the bilinear space over F with scalar product given by M has determinant 1),

(ii) M is not congruent to A B with a square A of odd size. ⊕ Docovićand- Szechtman [4] also proved that if F consists of more than 2 elements and its characteristic is 2 then M Mn(F) satisfies (1) if and only if ∈ M is not congruent to A B in which A is a singular Jordan block of odd size. ⊕ (Clearly, each M Mn(F) satisfies (1) if F has only 2 elements.) Coakley, ∈ Dopico, and Johnson [3, Corollary 4.10] gave another proof of Theorem 1 for real and complex matrices only: they used Thompson’s canonical pairs of symmetric and skew-symmetric matrices for congruence [14]. We give another proof of Theorem 1 using our canonical matrices for congruence [9, 11]. For the complex field, pairs of canonical forms of 8 different types are required in [3]; our canonical forms are of only three simple types (14). Our approach to Theorem 1 is via canonical forms of matrices; the approach in [4] is via decompositions of bilinear spaces. Following [3], we denote by Ξn(F) the set of all M Mn(F) that satisfy ∈ (1). A computation reveals that Ξn(F) is closed under congruence, that is,

M Ξn(F) and M congruent to N imply N Ξn(F). (3) ∈ ∈ The implication (i) (ii) of Theorem 1 is easy to establish: let M be ⇒ congruent to N = A B, in which A Mr(F) and r is odd. If S := T ⊕ ∈ r ( Ir) In−r, then S NS = N and det S =( 1) = 1, and so N / Ξn(F). − ⊕ − − ∈ It follows from (3) that M / Ξn(F). ∈ The implication (ii) (i) is not so easy to establish. It is proved in Section 3. In the rest of this section⇒ and in Section 2 we discuss some consequences of Theorem 1. The ﬁrst is

Corollary 1. Let F be a ﬁeld of characteristic not 2. If n is odd then Ξn(F) is empty. M Ξ (F) if and only if M is not symmetric. ∈ 2

3 Indeed, Theorem 1 ensures that M / Ξ2(F) if and only if M is congruent to [a] [b] for some a, b F, and this happens∈ if and only if M is symmetric. ⊕ ∈ In all matrix pairs that we consider, both matrices are over F and have the same size. Two matrix pairs (A, B) and (C,D) are equivalent if there exist nonsingular matrices R and S over F such that

R(A, B)S := (RAS, RBS)=(C,D).

A direct sum of pairs (A, B) and (C,D) is the pair

(A, B) (C,D) := (A C, B D) ⊕ ⊕ ⊕ The adjoint of (A, B) is the pair (BT , AT ); thus, (A, B) is selfadjoint if A is square and A = BT . For notational convenience, we write

M −T := (M −1)T .

We say that (A, B) is a direct summand of (M, N) for equivalence if (M, N) is equivalent to (A, B) (C,D) for some (C,D). A square matrix A ⊕ is a direct summand of M for congruence (respectively, similarity) if M is congruent (respectively, similar) to A B for some B. The criterion (ii) in Theorem 1 uses⊕ the relation of matrix congruence; one must solve a system of quadratic equations to check that two matrices are congruent. The criteria (iii) and (iv) in the following theorem can be more convenient to use: one must solve only a system of linear equations to check that two matrices are equivalent or similar. In Section 2 we show that Theorem 1 implies

Theorem 2. Let M be an n n matrix over a ﬁeld F of characteristic diﬀerent from 2. The following× conditions are equivalent:

(i) M / Ξn(F); ∈ (ii) M has a direct summand for congruence that has odd size;

(iii) (M T , M) has a direct summand (A, B) for equivalence, in which A and B are r r matrices and r is odd. × (iv) (in the case of nonsingular M) M −T M has a direct summand for similarity that has odd size.

4 For each positive integer r, deﬁne the (r 1)-by-r matrices − 10 0 01 0 . . . . Fr :=  .. ..  , Gr :=  .. ..  , (4) 0 10 0 01     and the r-by-r matrices

0 · · λ 0  1 ·  · · 1 λ  1 1 · Jr(λ) := . . , Γr :=  − −  . (5) .. ..  1 1      0 1 λ  1 1     − −    11 0    Note that −T r+1 Γ Γr is similar to Jr(( 1) ) (6) r − since

T ...... · 1 2  · ·  ∗ 1 1 1 1  ..  −T r+1 − − − − r+1 1 . Γr Γr =( 1)  1 1 1  Γr =( 1) . . −  · −  .. 2  1 1    − −  0 1  1 0      Explicit direct summands in the conditions (ii)–(iv) of Theorem 2 are given in the following theorem. Theorem 3. Let M be an n n matrix over a ﬁeld F of characteristic diﬀerent from 2. The following× conditions are equivalent:

(i) M / Ξn(F); ∈ (ii) M has a direct summand for congruence that is either

−T – a nonsingular matrix Q such that Q Q is similar to Jr(1) with odd r (if F is algebraically closed, then we can take Q to be Γr since any such Q is congruent to Γr), or

– Js(0) with odd s.

5 T (iii) (M , M) has a direct summand for equivalence that is either (Ir,Jr(1)) with odd r, or (Ft,Gt) with any t. (iv) (in the case of nonsingular M) M −T M has a direct summand for similarity that is Jr(1) with odd r. In the following section we deduce Theorems 2 and 3 from Theorem 1 and give an algorithm to determine if M Ξn(F). In Section 3 we prove Theorem 1. ∈

2 Theorem 1 implies Theorems 2 and 3

Theorem 3 gives three criteria for M / Ξn(F) that involve direct summands ∈ of M for congruence, direct summands of (M T , M) for equivalence, and direct summands of M −T M for similarity. In this section we deduce these criteria from Theorem 1. For this purpose, we recall the canonical form of square matrices M for congruence over F given in [11, Theorem 3], and derive canonical forms of selfadjoint pairs (M T , M) for equivalence and canonical forms of cosquares M −T M for similarity. Then we establish conditions on these canonical forms under which M / Ξn(F). ∈ 2.1 Canonical form of a square matrix for congruence Every square matrix A over a ﬁeld F of characteristic diﬀerent from 2 is similar to a direct sum, uniquely determined up to permutation of summands, of Frobenius blocks 0 0 cm . −. 1 .. .  Φpl = . , (7)  .. 0 c   − 2  0 1 c   − 1  in which l m m−1 p(x) = x + c x + + cm 1 ··· is an integer power of a polynomial

s s−1 p(x)= x + a x + + as (8) 1 ··· that is irreducible over F. This direct sum is the Frobenius canonical form of A; sometimes it is called the rational canonical form (see [2, Section 6]).

6 A Frobenius block has no direct summand under similarity other than itself, i.e., it is indecomposable under similarity. Also, the Frobenius block Φ(x−λ)m is similar to the Jordan block Jm(λ). If p(0) = as = 0 in (8), we deﬁne 6 ∨ −1 s −1 s −1 p (x) := a (1 + a x + + asx )= p(0) x p(x ) (9) s 1 ··· and observe that

(p(x)l)∨ = p(0)−lxslp(x−1)l =(p(0)−1xsp(x−1))l =(p∨(x))l. (10)

The matrix A−T A is the cosquare of a nonsingular matrix A. If two nonsingular matrices are congruent, then their cosquares are similar because

(ST AS)−T (ST AS)= S−1A−T AS. (11)

If Φ is a cosquare, we choose a matrix A such that A−T A = Φ and write √T Φ := A (a cosquare root of Φ).

Lemma 1. Let p(x) be an irreducible polynomial of the form (8) and let Φpl be an m m Frobenius block (7). Then ×

(a) Φpl is a cosquare if and only if

p(x) = x, p(x) = x +( 1)m+1, and p(x)= p∨(x). (12) 6 6 −

(b) If Φ l is a cosquare and m is odd, then p(x)= x 1. p − T Proof. The conditions in (a) and an explicit form of Φpl were established in [11, Theorem 7]; see [9, Lemma 2.3] for a more detailedp proof. ∨ −1 (b) By (12), p(x)= p (x). Therefore, as = a , so as = ε = 1 and s ± 2k+1 2k k+1 k p(x)= x + a x + + akx + akεx + + a εx + ε. 1 ··· ··· 1 Observe that p( ε)=0. But p(x) is irreducible, so s = 1 and p(x)= x + ε. − By (12) again, ε = 1. Therefore, p(x)= x 1. 6 − Deﬁne the skew sum of two matrices: 0 B [A B] := . A 0

7 Theorem 4. Let M be a square matrix over a ﬁeld F of characteristic different from 2. Then (a) M is congruent to a direct sum of matrices of the form

[Φpl Im], Q, Js(0), (13)

in which Φpl is an m m Frobenius block that is not a cosquare, Q is nonsingular and Q−T Q× is similar to a Frobenius block, and s is odd.

(b) M / Ξn(F) if and only if M has a direct summand for congruence that is either∈

−T – a nonsingular matrix Q such that Q Q is similar to Jr(1) with odd r, or

– Js(0) with odd s. Proof. (a) This statement is the existence part of Theorem 3 in [11] (also presented in [9, Theorem 2.2]), in which a canonical form of a matrix for congruence over F is given up to classification of Hermitian forms over fi- nite extensions of F. The canonical block J2m(0) is used in [11] instead of [Jm(0) Im], but the proof of Theorem 3 in [11] shows that these two matrices are congruent. (b) The “if” implication follows directly from Theorem 1. Let us prove the “only if” implication. If M / Ξn(F), Theorem 1 ensures that M is ∈ congruent to A B, in which A is square and has odd size. Part (a) ensures that A is congruent⊕ to a direct sum of matrices of the form (13), not all of which have even size. Thus, A (and hence also M) has a direct summand for congruence that is either Js(0) with s odd, or a nonsingular matrix Q of odd −T size such that Q Q is similar to a Frobenius block Φpl of odd size. Lemma −T 1 ensures that p(x) = x 1, so Q Q is similar to Φ x− r , which is similar − ( 1) to Jr(1). If F is algebraically closed, then Theorem 4 can be simplified as follows. Theorem 5. Let M be a square matrix over an algebraically closed field of characteristic different from 2. Then (a) M is congruent to a direct sum of matrices of the form

[Jm(λ) Im], Γr, Js(0), (14)

8 in which λ =( 1)m+1, each nonzero λ is determined up to replacement −1 6 − by λ , Γr is deﬁned in (5), and s is odd. This direct sum is uniquely determined by M, up to permutation of summands.

(b) M / Ξn(F) if and only if M has a direct summand for congruence of ∈ the form Γr with odd r or Js(0) with odd s.

Proof. (a) This canonical form for congruence was obtained in [9, Theorem 2.1(a)]; see also [6, 8]. (b) This statement follows from (a) and Theorem 1. The equivalence (i) (ii) in Theorem 3 follows from Theorems 4 and 5. (The equivalence (i) ⇔(ii) in Theorem 2 is another form of Theorem 1.) ⇔ 2.2 Canonical form of a selfadjoint matrix pair for equivalence Kronecker’s theorem for matrix pencils [5, Chapter 12] ensures that each matrix pair (A, B) over C is equivalent to a direct sum of pairs of the form

T T (Im,Jm(λ)), (Jr(0),Ir), (Fs,Gs), (Ft ,Gt ), in which Fs and Gs are deﬁned in (4). This direct sum is uniquely determined by (A, B), up to permutations of summands. Over a ﬁeld F of characteristic not 2, this canonical form with Frobenius blocks Φpl (see (7)) instead of Jordan blocks Jm(λ) can be constructed in two steps:

Use Van Dooren’s regularization algorithm [15] for matrix pencils • (which was extended to matrices of cycles of linear mappings in [13] and to matrices of bilinear forms in [7]) to transform (A, B) to an equivalent pair that is a direct sum of the regular part (Ik, R) with nonsingular R T T and canonical pairs of the form (Jr(0),Ir), (Fs,Gs), and (Ft ,Gt ).

Reduce R to a direct sum of Frobenius blocks Φpl by a similarity • transformation S−1RS; the corresponding similarity transformation −1 −1 S (Ik, R)S = (Ik,S RS) decomposes the regular part into a direct sum of canonical blocks (Im, Φpl ). Theorem 6. Let M be a square matrix over a ﬁeld F of characteristic different from 2.

9 (a) The selfadjoint pair (M T , M) is equivalent to a direct sum of selfadjoint pairs of the form

T T T T T ([Im Φpl ], [Φpl Im]), ( Φqr , Φqr ), (Js(0) ,Js(0)), (15) p p in which Φpl is an m m Frobenius block that is not a cosquare, Φqr is a Frobenius block that× is a cosquare, and s is odd. This direct sum is uniquely determined by M, up to permutations of direct summands and replacement, for each Φpl , of any number of summands of the form T T ∨ ([Im Φpl ], [Φpl Im]) by ([Im Φql ], [Φql Im]), in which q(x) := p (x) is deﬁned in (9).

(b) The following three conditions are equivalent:

(i) M / Ξn(F); ∈ (ii) (M T , M) has a selfadjoint direct summand for equivalence of the T T form (Γr , Γr) with odd r, or (Js(0) ,Js(0)) with odd s; (iii) (M T , M) has a direct summand for equivalence of the form (Ir,Jr(1)) with odd r, or (Ft,Gt) with any t.

Proof. Let M be a square matrix over a field F of characteristic different from 2. (a) By Theorem 4(a), M is congruent to a direct sum N of matrices of the form (13). Hence, (M T , M) is equivalent to (N T , N), a direct sum of pairs of the form (15). Uniqueness of this direct sum follows from the uniqueness assertion in Kronecker’s theorem and the following four equivalences: T 1. ([Im Φ l ], [Φ l Im]) is equivalent to (Im, Φ l ) (Im, Φ ∨ l ) for p(x) p(x) p(x) ⊕ p (x) each irreducible polynomial p(x) = x. T 6 2. ([Im Jm(0) ], [Jm(0) Im]) is equivalent to (Im,Jm(0)) (Jm(0),Im). T T T ⊕ 3. ( Φqr , Φqr ) is equivalent to (I, Φqr ). T T T 4. (Jp2t−1(0) p,J2t−1(0)) is equivalent to (Ft ,Gt ) (Gt, Ft). T⊕ To verify the first equivalence, observe that (Φp(x)l ,Im) is equivalent to (Im, Φp∨(x)l ) because

−T ∨ l Φp(x)l is similar to Φp (x) (16)

10 for each nonsingular m m Frobenius block Φ := Φp(x)l . The similarity (16) × −T follows from the fact that the characteristic polynomials of Φ and Φp∨(x)l are equal:

−1 −1 χ −T (x) = det(xI Φ ) = det(( Φ )(I xΦ)) Φ − − − = det( Φ−1) xm det(x−1I Φ) = χ∨ (x)=(p(x)l)∨, − · · − Φ which equals p∨(x)l by (10). The second equivalence is obvious. To verify the third equivalence, compute

−T T T T T Φqr ( Φqr , Φqr )I =(I, Φqr ). p p p The matrix pairs in the fourth equivalence are permutationally equivalent. (b) “(i) (ii)” Suppose that M / Ξn(F). By Theorem 4(b), M has a di- ⇒ ∈ −T rect summand Q for congruence such that Q Q is similar to Jr(1) with odd T T r, or a direct summand Js(0) with odd s. Then (Q , Q)or(Js(0) ,Js(0)) is a direct summand of (M T , M) for equivalence. The pair (QT , Q) is equivalent T −T −T to (Γr , Γr) since Q Q and Γr Γr are similar (they are similar to Jr(1) by (6)) and because

−1 −T −T T −1 −T T T S Q QS =Γ Γr = Γ S Q (Q , Q)S = (Γ , Γr). r ⇒ r r T “(ii) (iii)” To prove this implication, observe that (Γr , Γr) with odd r ⇒ −T is equivalent to (Ir, Γr Γr), which is equivalent to (Ir,Jr(1)) by (6), and [9, p. 213] ensures that

T T T (J t− (0) ,J t− (0)) is equivalent to (Ft,Gt) (G , F ). (17) 2 1 2 1 ⊕ t t “(iii) (i)” Assume the assertion in (iii). By Theorem 4(a), M is congru- ⇒ T ent to a direct sum N = iNi of matrices of the form (13). Then (M , M) T ⊕ T is equivalent to (N , N)= i(Ni , Ni). By (iii) and the uniqueness assertion ⊕ T in Kronecker’s theorem, some (Ni , Ni) has a direct summand for equivalence of the form (Ir,Jr(1)) with odd r or (Ft,Gt) with any t.

Suppose that the direct summand is (Ir,Jr(1)) with odd r. Since Ni is • one of the matrices (13) and Jr(1) with odd r is a cosquare by (12), it −T follows that Ni = Q and Q Q is similar to Jr(1).

Suppose that the direct summand is (Ft,Gt). Since Ni is one of the • matrices (13), (17) ensures that Ni = J2t−1(0).

11 In both the preceding cases, Ni has odd size, so Theorem 1 ensures that M / Ξn(F). ∈ The equivalences (i) (iii) in Theorems 2 and 3 follow from Theorem 6. ⇔ 2.3 Canonical form of a cosquare for similarity Theorem 7. Let M be a nonsingular matrix over a ﬁeld F of characteristic diﬀerent from 2.

(a) The cosquare M −T M is similar to a direct sum of cosquares

−T Φ l Φ l , Φqr , (18) p ⊕ p

in which Φpl is a nonsingular Frobenius block that is not a cosquare and Φqr is a Frobenius block that is a cosquare. This direct sum is uniquely determined by M, up to permutation of direct summands and replacement, for each Φpl , of any number of summands of the form −T −T ∨ Φ l Φ by Φ l Φ , in which q(x) := p (x) is deﬁned in (9). p ⊕ pl q ⊕ ql −T (b) M / Ξn(F) if and only if M M has a direct summand for similarity ∈ of the form Jr(1) with odd r.

Proof. (a) The existence of this direct sum follows from Theorem 4(a) since M is congruent to a direct sum of nonsingular matrices [Φpl Im] and Q (see (13)); the matrices (18) are their cosquares. The uniqueness assertion follows from uniqueness of the Frobenius canonical form and (16). (b) By Theorem 6(b) and because M is nonsingular, M / Ξn(F) if and T ∈ only if (M , M) has a direct summand for equivalence of the form (Ir,Jr(1)) T −T with odd r. This implies (b) since (M , M) is equivalent to (In, M M). The equivalences (i) (iv) in Theorems 2 and 3 follow from Theorem 7. ⇔ 2.4 An algorithm

The following simple condition is sufficient to ensure that M Ξn(F). ∈ Lemma 2 ([3, Theorem 2.3] for F = R or C). Let F be a field of characteristic different from 2. If M Mn(F) and if its skew-symmetric part Mw =(M T ∈ − M )/2 is nonsingular, then M Ξn(F). ∈

12 Proof. Since Mw is skew-symmetric and nonsingular, there exists a non- T singular C such that Mw = C Z2mC, in which Z2m is deﬁned in (2). If ST MS = M, then

T −1 T −1 S MwS = Mw, (CSC ) Z2m(CSC )= Z2m, and so CSC−1 is symplectic. By [1, Theorem 3.25], det CSC−1 = 1, which implies that det S = 1.

Independent of any condition on Mw, one can use the regularization algorithm described in [7] to reduce M by a sequence of congruences (simple row and column operations) to the form

B Jn1 (0) Jn (0), B nonsingular and 1 6 n 6 6 np. (19) ⊕ ⊕···⊕ p 1 ··· Of course, the singular blocks are absent and B = M if M is nonsingular. According to Theorem 7(b), the only information needed about B in (19) −T is whether it has any Jordan blocks Jr(1) with odd r. Let rk = rank(B B k −T − I) and set r = n. For each k =1,...,n, B B has rk− rk blocks Jj(1) of 0 1 − all sizes j k and exactly (r2k r2k+1) (r2k+1 r2k+2)= r2k 2r2k+1 +r2k+2 ≥ − − − n−1 − blocks of the form J2k+1(1) for each k =0, 1,..., [ 2 ]. The preceding observations lead to the following algorithm to determine whether a given M Mn(F) is in Ξn(F): ∈ T 1. If M M is nonsingular, then stop: M Ξn(F). − ∈ 2. If M is singular, use the regularization algorithm [7] to determine a direct sum of the form (19) to which M is congruent, and examine the singular block sizes nj. If any nj is odd, then stop: M / Ξn(F). ∈

3. If M is nonsingular or if all nj are even, then M Ξn(F) if and only if n−1 ∈ r k 2r k + r k = 0 for all k =0, 1,..., [ ]. 2 − 2 +1 2 +2 2 T Notice that if M M is nonsingular, then (a) no nj is odd since T − T Jr(0) Jr(0) is singular for every odd r, (b) B B is nonsingular, and (c) −−T −T T − rank(B B I) = rank(B (B B )) = n, so rk = n for all k = 1, 2,... − − and r k 2r k + r k = 0 for all k =0, 1,.... 2 − 2 +1 2 +2

13 3 Proof of Theorem 1

The implication (i) (ii) of Theorem 1 was established in Section 1. In this section we prove⇒ the remaining implication (ii) (i): we take any M ⇒ ∈ Mn(F) that has no direct summands for congruence of odd size, and show that M Ξn(F). We continue to assume, as in Theorem 1, that F is a ﬁeld ∈ of characteristic diﬀerent from 2. By (3) and Theorem 4(a), we can suppose that M is a direct sum of matrices of even sizes of the form [Φpl Im] and Q; see (13). Rearranging summands, we represent M in the form

M = M ′ M ′′, M ′ is n′ n′, M ′′ is n′′ n′′, (20) ⊕ × × in which

′ (α) M is the direct sum of all summands of the form [Φ(x−1)m Im] (m is even by Lemma 1(a)), and

(β) M ′′ is the direct sum of the other summands; they have the form [Φpl Im] with p(x) = x 1 and Q of even size, in which Φpl is an 6 − − m m Frobenius block that is not a cosquare and Q T Q is similar to a Frobenius× block.

Step 1: Show that for each nonsingular S,

ST MS = M = S = S′ S′′, S′ is n′ n′, S′′ is n′′ n′′. (21) ⇒ ⊕ × ×

If ST MS = M, then ST (M T , M)S = (M T , M), and so with R := S−T we have (M T , M)S = R(M T , M). (22) To prove (21), we prove a more general assertion: (22) implies that

S = S′ S′′, R = R′ R′′, S′, R′ are n′ n′, S′′, R′′ are n′′ n′′. (23) ⊕ ⊕ × × Using Theorem 5(a), we reduce M ′ and M ′′ in (20) by congruence transformations over the algebraic closure F of F to direct sums of matrices of the form [Jm(1) Im] and, respectively, of the form [Jm(λ) Im] with λ = 1 and 6 Γr with even r. Then

14 (M ′T , M ′) is equivalent over F to a direct sum of pairs of the form • (Im,Jm(1)) (Jm(1),Im), and ⊕ (M ′′T , M ′′) is equivalent over F to a direct sum of pairs of the form • T (Im,Jm(λ)) (Jm(λ),Im) with 1 = λ F and (Γ , Γr) with even r. ⊕ 6 ∈ r T The pair (Jm(1),Im) is equivalent to (Im,Jm(1)). The pair (Γr , Γr) is equiv- −T alent to (Ir, Γr Γr), which is equivalent to (Ir,Jr( 1)) by (6) since r is even. Thus, −

(α′) (M ′T , M ′) is equivalent to a direct sum of pairs of that are of the form (Im,Jm(1)), and

(β′) (M ′′T , M ′′) is equivalent to a direct sum of pairs that are either of the form (Im,Jm(λ)) with λ =1 or of the form (Jm(0),Im). 6 We choose γ F, γ = 1, such that M ′′T + γM ′′ is nonsingular (if M ′′ ∈ 6 − ′′ is nonsingular, then we may take γ = 0; if M is singular, then we may choose any γ = 0, 1 such that (M T , M) has no direct summands of the 6 −1 − form (Im,Jm( γ )). − Then (22) implies that

(M T + γM,M)S = R(M T + γM,M).

T T −1 The pair (M + γM,M) is equivalent to (In, (M + γM) M), whose Kronecker canonical pair has the form

′ ′′ (In, N) := (In′ , N ) (In′′ , N ), ⊕ in which (α′) and (β′) ensure that

(α′′) N ′ (of size n′ n′) is a direct sum of Jordan blocks with eigenvalue (1 + γ)−1, and×

(β′′) N ′′ (of size n′′ n′′) is a direct sum of Jordan blocks with eigenvalues × − distinct from (1 + γ) 1.

′′ ′′ If (In, N)S˜ = R˜(In, N), then S˜ = R˜, NS˜ = SN˜ , and(α ) and (β ) ensure ′ ′′ ′ ′ ′ ′′ ′′ ′′ that S˜ = S˜ S˜ , in which S˜ is n n and S˜ is n n . Since (In, N) is obtained from⊕ (M T , M) by transformations× within (M×′T , M ′) and within (M ′′T , M ′′), (22) implies (23). This proves (21).

15 Since det S = det S′ det S′′, it remains to prove that ′ ′′ M Ξn′ (F), M Ξn′′ (F). ∈ ∈ ′′ Step 2: Show that M Ξn′′ (F). ∈ ′′ ′′ ′′T By Lemma 2, it suﬃces to show that 2Mw = M M is nonsingular. This assertion is correct since (β) ensures that the matrix− M ′′ is a direct sum of matrices of the form [Φ l Im] with p(x) = x 1 and Q of even size, and p 6 − for each summand of the form [Φ l Im], • p T 0 Im Φpl [Φpl Im]w = − Φ l Im 0 p − is nonsingular since 1 is not an eigenvalue of Φpl ;

T T −T for each summand of the form Q, Q Q = Q (Q Q Ir) is non- • −T − − singular since Q Q is similar to a Frobenius block Φpl of even size, in which (12) ensures that p(x) = x 1, and so 1 is not an eigenvalue of − 6 − Q T Q. ′ Step 3: Show that M Ξn′ (F). By (α), M ′ is a direct∈ sum of matrices of the form

[Φ(x−1)m Im], m is even, (24) in which Φ(x−1)m is a Frobenius block that is not a cosquare; (12) ensures that m is even. −1 Since C Φ(x−1)m C = Jm(1) for some nonsingular C, each summand [Φ(x−1)m Im] is congruent to

T 0 Im C 0 0 Im C 0 = −1 −T , Jm(1) 0 0 C Φ(x−1)m 0 0 C which is congruent to 0 I˜ I˜ 0 0 I I˜ 0 m = m m m , J˜m(1) 0 0 ImJm(1) 0 0 Im in which 0 1 0 1  1 1 I˜ := , J˜ (1) := (m-by-m). m  · ·  m 1· 0  · · · ·    11· · 0   16 The matrix [J˜m(1) I˜m] is congruent via a permutation matrix to

0 K2  K2 L2  0 1 0 0 , in which K := , L := . (25) 2 1 0 2 1 0  · · · ·  K ·L · 0   2 2  We have proved that [Φ(x−1)m Im] is congruent to (25). Respectively,

[Φ x− m Im] [Φ x− m Im] (r summands) ( 1) ⊕···⊕ ( 1) is congruent to

0 Kr  Kr Lr  2 Am,r := (m blocks), (26)  · · · ·  K ·L · 0   r r  in which 0 Ir 0 0r Kr := , Lr := . Ir 0 Ir 0 Therefore, M ′ is congruent to some matrix

N = Am1,r1 Am2,r2 Am ,r , m > m > > mt, ⊕ ⊕···⊕ t t 1 2 ··· m in which ri is the number of summands [Φ(x−1) i Imi ] of size 2mi in the ′ direct sum M . In view of (3), it suﬃces to prove that N Ξn′ (F). ∈ If ST NS = N, (27) then (11) implies that N −T NS = SN −T N, (28) in which −T Am1,r1 Am1,r1 0 −T . N N =  ..  . (29) −T  0 Am ,r Amt,rt   t t  Since T ... L Kr ∗ ∗ − ri i  . T  . L Kr − · · ri i A 1 = · − , mi,ri  · Kri   ∗ T · ·   Lri ·  − · ·   Kr 0   i  17 we have

I r Hr ... 2 i i ∗ ∗  .. .  I2ri Hri . . −T . Iri 0 A Am ,r = . , Hr := ; (30) mi,ri i i  I2ri .  i   0 Iri  . ∗  −  .. H   ri   0 I   2ri  the stars denote unspeciﬁed blocks. Partition S in (28) into t2 blocks

S11 ... S1t . . . S =  . .. .  , Sij is 2miri 2mjrj, × St1 ... Stt    conformally to the partition (29), then partition each block Sij into subblocks of size 2ri 2rj conformally to the partition (30) of the diagonal blocks of (29). Equating× the corresponding blocks in the matrix equation (28) (much as in Gantmacher’s description of all matrices commuting with a Jordan matrix, [5, Chapter VIII, 2]), we ﬁnd that § all diagonal blocks of S have the form •

Ci H ∗  Ci  .. H Sii =  .  , Ci := Hri CiHri ,    Ci   H   0 Ci    (the number of diagonal blocks is even by (24)), and

all oﬀ-diagonal blocks Sij have the form • ... ∗ . ∗. ...  .. . ∗ . ∗. if i < j,  .. . if i > j,    ∗ 0  0   ∗  

18 in which the stars denote unspeciﬁed subblocks.1 For example, if

N = A ,r1 A ,r2 A ,r3 6 ⊕ 4 ⊕ 2 0 Kr1

 Kr1 Lr1  0 Kr2 Kr1 Lr1  Kr2 Lr2  0 Kr3 =   ,  Kr1 Lr1 ⊕ Kr2 Lr2 ⊕Kr3 Lr3      Kr1 Lr1  Kr2 Lr2 0      Kr1 Lr1 0    then

C1 ∗H ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗  C1  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ C1  ∗ ∗ ∗ ∗ ∗   CH   1 ∗ ∗ ∗   C1   ∗   CH  S =  1  ,  C2   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗   CH   ∗ ∗ ∗ 2 ∗ ∗ ∗   C   ∗ ∗ 2 ∗   CH   ∗ 2   C   ∗ ∗ ∗ ∗ 3 ∗   CH   ∗ ∗ 3  in which

H H H C1 := Hr1 C1Hr1 , C2 := Hr2 C2Hr2 , C3 := Hr3 C3Hr3 . Now focus on equation (27). The subblock at the upper right of the ith diagonal block Ami,ri of N is Kri ; see (26). Let us prove that the corresponding T T H subblock of S NS is Ci Kri Ci ; that is,

T H Ci Kri Ci = Kri . (31) 1 Each Jordan matrix J is permutation similar to a Weyr matrix WJ and all matrices commuting with WJ are block triangular; see [12, Section 1.3]. If we reduce the matrix (29) by simultaneous permutations of rows and columns to its Weyr form, then the same permutations reduce S to block triangular form.

19 Multiplying the ﬁrst horizontal substrip of the ith strip of ST by N, we obtain

T (0 ... 0 ... 0 ... 0 0 ... 0 C Kr 0 ... 0 ... 0 ... 0); ∗| | ∗| i i | | | multiplying it by the last vertical substrip of the ith vertical strip of S, we T H H obtain Ci Kri Ci , which proves (31). Thus, det Ci det Ci = 1. But

det S = det C det CH det C det CH det C det CH 1 1 ··· 1 1 2 2 ··· Therefore, det S = 1, which completes the proof of Theorem 1.

Acknowledgment

The authors are very grateful to the referee for valuable remarks and sug- gestions, and to Professor F. Szechtman for informing us of the important paper [4].

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